Direct Noise Computation of Linear and Nonlinear Rotor-Stator Interaction Modes in Transonic Cascades

摘要

A surface interpolation scheme that has a formal fourth-order precision in space and has no amplification in the frequency domain is used to directly compute the linear and nonlinear rotor-stator interaction modes in transonic cascades. The time-dependent and compressible Euler equations are numerically solved using a finite volume discretization where the fluxes are computed using the proposed surface interpolation scheme, while the time marching process is achieved using a third-order Runge-Kutta scheme. The immersed boundary method is based on a discrete forcing approach where the boundary conditions are directly imposed in the control volumes that contain the immersed boundary points, resulting in a sharp representation of the static and moving solid boundaries. For the simulated cases, those boundaries correspond to the tip geometry of the rotor blades and stator vanes of the Advanced Noise Control Fan (ANCF). The simulated operational conditions are theoretical and off-design, since the imposed Mach number for the rotor is 0.9564. Two cases are simulated, the rotor alone and the rotor-stator interaction. For the first case, nonlinear modes are generated but all of them are cut-off in the linear region, as predicted by the linear theory. For the second case, all of the generated modes in the interstage region are nonlinear, i.e., the amplitude of the pressure fluctuations are at least an order of magnitude grater than the value associated to the linear modes. The nonlinear modes due to the rotor-stator interaction are generated in the interstage region following the Tyler-Sofrin rule, and include modes that later 'will be cut-on or cut-off in the linear (upstream) region. In this linear region, only the cut-on modes are present and the numerical results are in excellent agreement with the theoretical results from the linear theory regarding the mode number, mode signal (positive or negative) and mode angle. This excellent agreement between the numerical results and the theoretical predictions for the linear modes and the fact that both, the nonlinear and linear modes, strictly follow the Tyler-Sofrin rule strongly suggest that the cut-on linear modes where initially generated as nonlinear modes in the interstage region.